M. Lochter Bundesamt fuer Sicherheit in der Informationstechnik (BSI) J. Merkle secunet Security Networks AG June 2007 ECC Brainpool Standard Curves and Curve Generation draft-lochter-pkix-brainpool-ecc-00.txt Status of this Memo By submitting this Internet-Draft, each author represents that any applicable patent or other IPR claims of which he or she is aware have been or will be disclosed, and any of which he or she becomes aware will be disclosed, in accordance with Section 6 of BCP 79. This document may not be modified, and derivative works of it may not be created, except to publish it as an RFC and to translate it into languages other than English. This document may only be posted in an Internet-Draft. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet-Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." The list of current Internet-Drafts can be accessed at http://www.ietf.org/1id-abstracts.html The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html Abstract This RFC proposes several elliptic curve domain parameters over finite prime fields for use in cryptographic applications. The domain parameters are consistent with the relevant international standards, and can be used in X.509 certificates and CRLs, IKE, TLS, XML signatures, and all applications or protocols based on the cryptographic message syntax (CMS). 1 Introduction Although several standards for elliptic curves and domain parameters exist (e.g. [ANSI1], [NIST] or [SEC2]), some major issues have still not been addressed: - The generation of the prime p and the seed from which the curve parameters were derived is irreproducible, leaving out an essential part of the security analysis. - No proofs are provided that the proposed parameters do not belong to those classes of parameters which are susceptible to cryptanalytic attacks with sub-exponential complexity. - Recent research results seem to indicate a potential for new attacks on elliptic curve cryptosystems. At least for applications with highest security demands or under circumstances which complicate a change of parameters in response to new attacks, the inclusion of a corresponding security requirement for domain parameters (the class group condition, see section 2) is justified. - Some of the proposed subgroups have a non-trivial cofactor, which demands additional checks by cryptographic applications to prevent small subgroup attacks (see [ANSI1] or [SEC1]). - The domain parameters specified do not cover all bit lengths that correspond to the commonly used key lengths for symmetric cryptographic algorithms. In particular, there is no 512 bit curve defined but only one with 521 bit length, which may be disadvantageous for some implementations. Furthermore, many of the parameters specified by the existing standards are identical (see [SEC2] for a comparison). Thus, there is still a need for additional elliptic curve domain parameters which overcome the above limitations. 1.1 Scope and relation to other specifications This RFC specifies elliptic curve domain parameters over prime fields GF(p) with p having a length of 160, 192, 224, 256, 320, 384 and 512 bits. These parameters were generated in a pseudo-random yet completely systematic and reproducible way and have been verified to resist current cryptanalytic approaches. The parameters are compliant with ANSI X9.62 ([ANSI1]) and X9.63 [ANSI2], ISO/IEC 14888 [ISO1] and ISO/IEC 15946 [ISO2], ETSI TS 102176 ([ETSI]), as well as with the specifications of NIST ([NIST]), SecG ([SEC1] and [SEC2]) and IEEE ([IEEE]). Furthermore, this document identifies and explains the requirements for the parameters that have led to the methods for the generation and security validation of the parameters. Complementing information, including the pseudo-random generation methods for the parameters and the security proofs, are given in [EBP]. Finally, this RFC defines ASN.1 object identifiers for all elliptic curve domain parameters specified herein, e.g. for use in X.509 certificates. This document does neither address the cryptographic algorithms to be used with the specified parameters nor their application in other standards. However, it is consistent with the following RFCs and internet drafts which specify the usage of elliptic curve cryptography in protocols and applications: - [RFC 3278] for the cryptographic message syntax (CMS) - [RFC 3279] and [PKIX] for X.509 certificates and CRLs - [RFC 4050] for XML signatures - [RFC 4492] for TLS - [IPSEC] for IKE 2 Requirements on the elliptic curve domain parameters Throughout this memo let p > 3 be a prime and GF(p) a finite field (sometimes also referred to as Galois Field or F_p) with p elements. For given A and B with non-zero 4*A^3 + 27*B^2 mod p, the set of solutions (x,y) for the equation E: y^2 = x^3 + A*x + B mod p over GF(p) together with a neutral element O and well-defined laws for addition and inversion define a group - the elliptic curve E(GF(p)). Typically, for cryptographic applications, an element G of prime order q is chosen. A comprehensive introduction to elliptic curves and their cryptographic applications can be found in [BSS]. Note 1: We choose {0,...,p-1} as a set of representatives for the elements of GF(p). This choice induces a natural ordering on GF(p). 2.1 Security Requirements. The following security requirements are either motivated by known cryptographic analysis or aim to enhance trust in the recommended curves. 1. Immunity to attacks using the Weil- or Tate-Pairing. These attacks allow the embedding of the cyclic subgroup generated by G into the group of units of a degree-l extension GF(p^l) of GF(p), where sub-exponential attacks on the discrete logarithm problem (DLP) exist. Here we have l = min{t | q divides p^t-1}, i.e. l is the order of p mod q. By Fermat's little theorem, l divides q-1. We require (q-1)/l < 100, which means that l is close to the maximum possible value. This requirement is considerably stronger than those of [SEC2] and [ANSI2] and also excludes supersingular curves, as those are the curves of order p+1. Detailed information on this requirement can be found in [BSS]. 2. The trace is not equal to one. Trace one curves (or anomalous curves) are curves with #E(GF(p)) = p. Satoh and Araki [SA], Semaev [Sem] and Smart [Sma] independently proposed efficient solutions to the elliptic curve discrete logarithm problem (ECDLP) on trace one curves. Note that these curves are also excluded by requirement 5 of section 2.2. 3. Large class number. The class number of the maximal order of the endomorphism ring End(E) of E is larger than 10^7. Generally, E cannot be "lifted" to a curve E' over an algebraic number field L with End(E) = End(E’) unless the degree of L over the rationals is larger than the class number of End(E). Although there are no efficient attacks exploiting a small class number, recent work ([JMV] and [HR]) also may be seen as argument for the class number condition. (See [EBP] for more details on class group computations.) This condition excludes curves that are generated by the well-known CM-method. 4. Prime group order. The group order #E(GF(p)) shall be a prime number in order to counter small-subgroup attacks ([HMV]). Therefore, all groups proposed in this RFC have cofactor 1. Note that curves with prime order have no point of order 2 and therefore no point with y-coordinate 0. 5. Verifiably pseudo-random. The elliptic curve domain parameters shall be generated in a pseudo-random manner using seeds that are generated in a systematic and comprehensive way. Our method of construction is explained in [EBP]. 6. Proof of security. For all curves a proof should be given that all security requirements are met. These proofs are provided in [EBP]. In [BG], attacks are described which apply to elliptic curve domain parameters where q-1 has a factor u in the order of q^(1/3)). However, the circumstances under which these attacks are applicable can be avoided in most applications. Therefore, no corresponding security requirement is stated here. However, it is highly recommended that developers verify the security of their implementations against this kind of attack. 2.2 Technical Requirements Commercial demands and experience with existing implementations lead to the following technical requirements for the elliptic curve domain parameters. 1. For each of the bit lengths 160, 192, 224, 256, 320, 384 and 512 one curve shall be proposed. This requirement follows from the need for curves providing different levels of security which are appropriate for the underlying symmetric algorithms. The existing standards specify a 521-bit curve instead of a 512-bit curve. 2. The prime number p shall be congruent 3 mod 4. This requirement allows efficient point compression: One method for the transmission of curve points P=(x,y) is to transmit only x and the least significant bit LSB(y) of y. Using the curve equation and p = 3 mod 4 we get (y^2)^(p+1)/4 = y*y^(p-1)/2 which is either y or -y by Fermat's little theorem and hence, y can be computed very efficiently. This requirement is not always met by the parameters defined in existing standards. 3. The curves shall be GF(p)-isomorphic to a "cryptographically good curve" (i.e. a curve that meets all security requirements defined in section 2.2) with A = -3 mod p. This property permits the use of the arithmetical advantages of curves with A = -3 mod p as shown by Brier and Joyce [BJ]. The requirement is fulfilled by a quadratic twist E' of the given curve E with a square in GF(p): If -3 = A*Z^4 mod p is solvable, then E and E': y^2 = x^3 + Z^4*A*x + Z^6*B mod p are GF(p)-isomorphic via the isomorphism F(x,y) := (x*Z^2, y*Z^3). Especially, #E(GF(p)) = #E'(GF(p)) and, most importantly, E and E' have the same algebraic structure and hence, offer the same level of security. Approximately half of the isomorphism classes of elliptic curves over GF(p) with p = 3 mod 4 contain a curve with A = -3 mod p. This constraint has also been used by [SEC2] and [NIST]. 4. The prime p must not be of any special form; this requirement is met by a verifiably pseudo-random generation of the parameters (see requirement 5 in section 2.1). Although parameters specified by existing standards do not meet this requirement, the need for such curves over (pseudo-)randomly chosen fields has already been foreseen by the Standards for Efficient Cryptography Group (SECG), see [SEC2]. 5. #E(GF(p)) < p. As a consequence of the Hasse-Weil-Theorem the number of points #E(GF(p)) may be greater than the characteristic p of the prime field GF(p). In some cases even the bit-length of #E(GF(p)) can exceed the bit-length of p. To avoid overruns in implementations we require that #E(GF(p)) < p. In order to thwart attacks on digital signature schemes, some authors propose to use q > p, but the attacks described e.g. in [BRS] appear infeasible in a well-designed PKI. 6. B shall be a non-square mod p. Otherwise, the compressed representations of the curve-points (0,0) and (0,X) with X being the square root of B with a least significant bit of 0 would be identical. As there are implementations of elliptic curves that encode the point at infinity as (0,0) we try to avoid ambiguities. Note that this condition is stable under quadratic twists as described in condition 3 above. Condition 6 makes the attack described in [G] impossible. It can therefore also be seen as a security requirement. This constraint has not been specified by existing standards. 3 Parameter specification In this section the elliptic curve domain parameters proposed are specified in the following way. For all curves an ID is given by which it can be referenced. p is the prime specifying the base field. A and B are the coefficients of the equation y^2 = x^3 + A*x + B mod p defining the elliptic curve E. G is the base point, i.e. a point in E of prime order. x and y are its x- and y-coordinates, respectively. q is the prime order of the group generated by G. h is the cofactor of G in E, i.e. #E(GF(p))/q. For the twisted curve E' the coefficient Z that defines the isomorphism F (see requirement 3 in section 2.2), the coefficients A' = Z^4*A mod p and B' = Z^6*B mod p of the curve equation and the base point G' are given. The methods for the generation of the parameters and complete security proofs regarding the security requirements specified in section 2.1 are given in [EBP]. 3.1 Parameters for 160 bit curves Curve-ID: brainpoolP160r1 p = E95E4A5F737059DC60DFC7AD95B3D8139515620F A = 340E7BE2A280EB74E2BE61BADA745D97E8F7C300 B = 1E589A8595423412134FAA2DBDEC95C8D8675E58 x = BED5AF16EA3F6A4F62938C4631EB5AF7BDBCDBC3 y = 1667CB477A1A8EC338F94741669C976316DA6321 q = E95E4A5F737059DC60DF5991D45029409E60FC09 h = 1 #Twisted curve Curve-ID: brainpoolP160t1 Z = 24DBFF5DEC9B986BBFE5295A29BFBAE45E0F5D0B A' = E95E4A5F737059DC60DFC7AD95B3D8139515620C B' = 7A556B6DAE535B7B51ED2C4D7DAA7A0B5C55F380 x = B199B13B9B34EFC1397E64BAEB05ACC265FF2378 y = ADD6718B7C7C1961F0991B842443772152C9E0AD q = E95E4A5F737059DC60DF5991D45029409E60FC09 h = 1 3.2 Parameters for 192 bit curves Curve-ID: brainpoolP192r1 p = C302F41D932A36CDA7A3463093D18DB78FCE476DE1A86297 A = 6A91174076B1E0E19C39C031FE8685C1CAE040E5C69A28EF B = 469A28EF7C28CCA3DC721D044F4496BCCA7EF4146FBF25C9 x = C0A0647EAAB6A48753B033C56CB0F0900A2F5C4853375FD6 y = 14B690866ABD5BB88B5F4828C1490002E6773FA2FA299B8F q = C302F41D932A36CDA7A3462F9E9E916B5BE8F1029AC4ACC1 h = 1 #Twisted curve Curve-ID: brainpoolP192t1 Z = 1B6F5CC8DB4DC7AF19458A9CB80DC2295E5EB9C3732104CB A' = C302F41D932A36CDA7A3463093D18DB78FCE476DE1A86294 B' = 13D56FFAEC78681E68F9DEB43B35BEC2FB68542E27897B79 x = 3AE9E58C82F63C30282E1FE7BBF43FA72C446AF6F4618129 y = 97E2C5667C2223A902AB5CA449D0084B7E5B3DE7CCC01C9 q = C302F41D932A36CDA7A3462F9E9E916B5BE8F1029AC4ACC1 h = 1 3.3 Parameters for 224 bit curves Curve-ID: brainpoolP224r1 p = D7C134AA264366862A18302575D1D787B09F075797DA89F57EC8C0FF A = 68A5E62CA9CE6C1C299803A6C1530B514E182AD8B0042A59CAD29F43 B = 2580F63CCFE44138870713B1A92369E33E2135D266DBB372386C400B x = D9029AD2C7E5CF4340823B2A87DC68C9E4CE3174C1E6EFDEE12C07D y = 58AA56F772C0726F24C6B89E4ECDAC24354B9E99CAA3F6D3761402CD q = D7C134AA264366862A18302575D0FB98D116BC4B6DDEBCA3A5A7939F h = 1 #Twisted curve Curve-ID: brainpoolP224t1 Z = 2DF271E14427A346910CF7A2E6CFA7B3F484E5C2CCE1C8B730E28B3F A' = D7C134AA264366862A18302575D1D787B09F075797DA89F57EC8C0FC B' = 4B337D934104CD7BEF271BF60CED1ED20DA14C08B3BB64F18A60888D x = 6AB1E344CE25FF3896424E7FFE14762ECB49F8928AC0C76029B4D580 y = 374E9F5143E568CD23F3F4D7C0D4B1E41C8CC0D1C6ABD5F1A46DB4C q = D7C134AA264366862A18302575D0FB98D116BC4B6DDEBCA3A5A7939F h = 1 3.4 Parameters for 256 bit curves Curve-ID: brainpoolP256r1 p = A9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5377 A = 7D5A0975FC2C3057EEF67530417AFFE7FB8055C126DC5C6CE94A4B44F330B5D9 B = 26DC5C6CE94A4B44F330B5D9BBD77CBF958416295CF7E1CE6BCCDC18FF8C07B6 x = 8BD2AEB9CB7E57CB2C4B482FFC81B7AFB9DE27E1E3BD23C23A4453BD9ACE3262 y = 547EF835C3DAC4FD97F8461A14611DC9C27745132DED8E545C1D54C72F046997 q = A9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7 h = 1 #Twisted curve Curve-ID: brainpoolP256t1 Z = 3E2D4BD9597B58639AE7AA669CAB9837CF5CF20A2C852D10F655668DFC150EF0 A' = A9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5374 B' = 662C61C430D84EA4FE66A7733D0B76B7BF93EBC4AF2F49256AE58101FEE92B04 x = A3E8EB3CC1CFE7B7732213B23A656149AFA142C47AAFBC2B79A191562E1305F4 y = 2D996C823439C56D7F7B22E14644417E69BCB6DE39D027001DABE8F35B25C9BE q = A9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7 h = 1 3.5 Parameters for 320 bit curves Curve-ID: brainpoolP320r1 p = D35E472036BC4FB7E13C785ED201E065F98FCFA6F6F40DEF4F92B9EC7893EC28FCD412 B1F1B32E27 A = 3EE30B568FBAB0F883CCEBD46D3F3BB8A2A73513F5EB79DA66190EB085FFA9F492F375 A97D860EB4 B = 520883949DFDBC42D3AD198640688A6FE13F41349554B49ACC31DCCD884539816F5EB4 AC8FB1F1A6 x = 43BD7E9AFB53D8B85289BCC48EE5BFE6F20137D10A087EB6E7871E2A10A599C710AF8D 0D39E20611 y = 14FDD05545EC1CC8AB4093247F77275E0743FFED117182EAA9C77877AAAC6AC7D35245 D1692E8EE1 q = D35E472036BC4FB7E13C785ED201E065F98FCFA5B68F12A32D482EC7EE8658E9869155 5B44C59311 h = 1 #Twisted curve Curve-ID: brainpoolP320t1 Z = 15F75CAF668077F7E85B42EB01F0A81FF56ECD6191D55CB82B7D861458A18FEFC3E5AB 7496F3C7B1 A' = D35E472036BC4FB7E13C785ED201E065F98FCFA6F6F40DEF4F92B9EC7893EC28FCD412 B1F1B32E24 B' = A7F561E038EB1ED560B3D147DB782013064C19F27ED27C6780AAF77FB8A547CEB5B4FE F422340353 x = 925BE9FB01AFC6FB4D3E7D4990010F813408AB106C4F09CB7EE07868CC136FFF3357F6 24A21BED52 y = 63BA3A7A27483EBF6671DBEF7ABB30EBEE084E58A0B077AD42A5A0989D1EE71B1B9BC0 455FB0D2C3 q = D35E472036BC4FB7E13C785ED201E065F98FCFA5B68F12A32D482EC7EE8658E9869155 5B44C59311 h = 1 3.6 Parameters for 384 bit curves Curve-ID: brainpoolP384r1 p = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B412B1DA197FB71123ACD3A7 29901D1A71874700133107EC53 A = 7BC382C63D8C150C3C72080ACE05AFA0C2BEA28E4FB22787139165EFBA91F90F8AA581 4A503AD4EB04A8C7DD22CE2826 B = 4A8C7DD22CE28268B39B55416F0447C2FB77DE107DCD2A62E880EA53EEB62D57CB4390 295DBC9943AB78696FA504C11 x = 1D1C64F068CF45FFA2A63A81B7C13F6B8847A3E77EF14FE3DB7FCAFE0CBD10E8E826E0 3436D646AAEF87B2E247D4AF1E y = 8ABE1D7520F9C2A45CB1EB8E95CFD55262B70B29FEEC5864E19C054FF99129280E4646 217791811142820341263C5315 q = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B31F166E6CAC0425A7CF3AB6 AF6B7FC3103B883202E9046565 h = 1 #Twisted curve Curve-ID: brainpoolP384t1 Z = 41DFE8DD399331F7166A66076734A89CD0D2BCDB7D068E44E1F378F41ECBAE97D2D63D BC87BCCDDCCC5DA39E8589291C A' = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B412B1DA197FB71123ACD3A7 29901D1A71874700133107EC50 B' = 7F519EADA7BDA81BD826DBA647910F8C4B9346ED8CCDC64E4B1ABD11756DCE1D2074AA 263B88805CED70355A33B471EE x = 18DE98B02DB9A306F2AFCD7235F72A819B80AB12EBD653172476FECD462AABFFC4FF19 1B946A5F54D8D0AA2F418808CC y = 25AB056962D30651A114AFD2755AD336747F93475B7A1FCA3B88F2B6A208CCFE469408 584DC2B2912675BF5B9E582928 q = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B31F166E6CAC0425A7CF3AB6 AF6B7FC3103B883202E9046565 h = 1 3.7 Parameters for 512 bit curves Curve-ID: brainpoolP512r1 p = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308717D4D9B 009BC66842AECDA12AE6A380E62881FF2F2D82C68528AA6056583A48F3 A = 7830A3318B603B89E2327145AC234CC594CBDD8D3DF91610A83441CAEA9863BC2DED5D 5AA8253AA10A2EF1C98B9AC8B57F1117A72BF2C7B9E7C1AC4D77FC94CA B = 3DF91610A83441CAEA9863BC2DED5D5AA8253AA10A2EF1C98B9AC8B57F1117A72BF2C7 B9E7C1AC4D77FC94CADC083E67984050B75EBAE5DD2809BD638016F723 x = 81AEE4BDD82ED9645A21322E9C4C6A9385ED9F70B5D916C1B43B62EEF4D0098EFF3B1F 78E2D0D48D50D1687B93B97D5F7C6D5047406A5E688B352209BCB9F822 y = 7DDE385D566332ECC0EABFA9CF7822FDF209F70024A57B1AA000C55B881F8111B2DCDE 494A5F485E5BCA4BD88A2763AED1CA2B2FA8F0540678CD1E0F3AD80892 q = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA70330870553E5C 414CA92619418661197FAC10471DB1D381085DDADDB58796829CA90069 h = 1 #Twisted curve Curve-ID: brainpoolP512t1 Z = 12EE58E6764838B69782136F0F2D3BA06E27695716054092E60A80BEDB212B64E585D9 0BCE13761F85C3F1D2A64E3BE8FEA2220F01EBA5EEB0F35DBD29D922AB A' = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308717D4D9B 009BC66842AECDA12AE6A380E62881FF2F2D82C68528AA6056583A48F0 B' = 7CBBBCF9441CFAB76E1890E46884EAE321F70C0BCB4981527897504BEC3E36A62BCDFA 2304976540F6450085F2DAE145C22553B465763689180EA2571867423E x = 640ECE5C12788717B9C1BA06CBC2A6FEBA85842458C56DDE9DB1758D39C0313D82BA51 735CDB3EA499AA77A7D6943A64F7A3F25FE26F06B51BAA2696FA9035DA y = 5B534BD595F5AF0FA2C892376C84ACE1BB4E3019B71634C01131159CAE03CEE9D99321 84BEEF216BD71DF2DADF86A627306ECFF96DBB8BACE198B61E00F8B332 q = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA70330870553E5C 414CA92619418661197FAC10471DB1D381085DDADDB58796829CA90069 h = 1 4 Object identifiers for the elliptic curve domain parameters The root of the tree for the object identifier of the domain parameters defined in this specification is given by ecStdCurvesAndGeneration OBJECT IDENTIFIER::= {iso(1) identifified-organization(3) teletrust(36) algorithm(3) signature-algorithm(3) ecSign(2) 8} The object identifier ellipticCurve represents the tree containing the object identifiers for each set of domain parameters specified in this RFC. It has the following value: ellipticCurve OBJECT IDENTIFIER ::= {ecStdCurvesAndGeneration 1} The tree for the domain parameters defined in this RFC is versionOne OBJECT IDENTIFIER ::= {ellipticCurve 1} The following object identifiers represent the domain parameters defined in this RFC: brainpoolP160r1 OBJECT IDENTIFIER ::= {versionOne 1} brainpoolP160t1 OBJECT IDENTIFIER ::= {versionOne 2} brainpoolP192r1 OBJECT IDENTIFIER ::= {versionOne 3} brainpoolP192t1 OBJECT IDENTIFIER ::= {versionOne 4} brainpoolP224r1 OBJECT IDENTIFIER ::= {versionOne 5} brainpoolP224t1 OBJECT IDENTIFIER ::= {versionOne 6} brainpoolP256r1 OBJECT IDENTIFIER ::= {versionOne 7} brainpoolP256t1 OBJECT IDENTIFIER ::= {versionOne 8} brainpoolP320r1 OBJECT IDENTIFIER ::= {versionOne 9} brainpoolP320t1 OBJECT IDENTIFIER ::= {versionOne 10} brainpoolP384r1 OBJECT IDENTIFIER ::= {versionOne 11} brainpoolP384t1 OBJECT IDENTIFIER ::= {versionOne 12} brainpoolP512r1 OBJECT IDENTIFIER ::= {versionOne 13} brainpoolP512t1 OBJECT IDENTIFIER ::= {versionOne 14} The ASN.1 syntax for elliptic curve domain parameters according to ANSI X9.62 [ANSI1] allows indicating whether a curve and base point have been generated verifiably at random or not. The parameters specified in section 3 have all been generated verifyably at random; however, the algorithms used for their generation deviate from those specified in ANSI X9.62. Consequently, applications following ANSI X9.62 will not be able to verify the randomness of the parameters. In order to avoid rejection of the paramaters, the ASN.1 encoding SHOULD NOT specify that the curve or base point has been generated verifiably at random. In particular, CAs SHOULD encode SpecifiedECDomain in the following way: - The field Version is set to ecdpVer1(1). - The field curve.seed is absent. - The field hash is absent. 5 Intellectual Property Rights The authors have no knowledge about any intellectual property rights which cover the usage of the domain parameters defined herein. However, readers should be aware that implementations based on these domain parameters may require use of inventions covered by patent rights. 6 References [ANSI1] ANSI X9.62-2005, Public Key Cryptography For The Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA). 2005. [ANSI2] ANSI X9.63-2001, Public Key Cryptography For The Financial Services Industry: Key Agreement and Key Transport Using The Elliptic Curve Cryptography. 2001. [BJ] E. Brier, M. Joyce, Fast multiplication on Elliptic Curves through Isogenies. In: M. Fossorier, T. Hoholdt, and A. Poli, eds., Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, LNCS 2643, Springer 2003. [BG] J. Brown and R. P. Gallant, The static Diffie-Hellman Problem, Technical Report CACR 2004-10, Centre for Applied Cryptographic Research, University of Waterloo, 2005, accessible via http://eprint.iacr.org/ [BRS] J. Bohli, S. Röhrich, R. Steinwandt. Key substitution attacks revisited: taking into account malicious signers. Preprint, 2004. [BSS] I. Blake, G. Seroussi, N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, 1999. [EBP] ECC Brainpool Standard Curves and Curve Generation, ECC Brainpool, October 2005, available at http://www.ecc-brainpool.org/download/ BP-Kurven-aktuell.pdf. [ETSI] ETSI TS 102 176-1, Algorithms and Parameters for Secure Electronic Signatures, Part 1: Hash functions and asymmetric algorithms. Version 1.2.1, 2005 [NIST] National Institute of Standards and Technology. FIPS PUB 186-2: Digital Signature Standard (DSS). 2000. [G] L. Goubin. A refined power-analysis-attack on Elliptic Curve Cryptosystems. In: Public-Key-Cryptography - PKC2003, Lecture Notes in Computer Science, 2567, Springer 2003. [HMV] D. Hankerson, A. Menezes, S. Vanstone. Guide to Elliptic Curve Cryptography. Springer 2004. [HR] Ming-Deh Huang and Wayne Raskind. Global methods for discrete logarithm problem III. Preprint 2005. [IPSEC] IPSec Working Group. IKE and IKEv2 Authentication Using ECDSA . Internet draft. July 2006. [ISO1] ISO/IEC 14888-3. Information technology — Security techniques — Digital signatures with appendix — Part 3: Discrete logarithm based mechanisms. Second Edition. 2006. [ISO2] ISO/IEC 15946-2. Information technology — Security techniques — Cryptographic techniques based on elliptic curves — Part 2: Digital signatures. First Edition. 2002. [JMV] D. Jao, S. D. Miller, R. Venkatesan, Ramanujan graphs and the random reducibility of discrete log on isogenous elliptic curves, IACR Cryptology ePrint Archive, 2004. [PKIX] PKIX Working Group. Additional Algorithms and Identifiers for use of Elliptic Curve Cryptography with PKIX , October 2006. [RFC2409] Harkins, D. and D. Carrel, "The Internet Key Exchange (IKE)", RFC 2409, November 1998. [RFC3278] Blake-Wilson, S., Brown, D., and P. Lambert, "Use of Elliptic Curve Cryptography (ECC) Algorithms in Cryptographic Message Syntax (CMS)", RFC 3278, April 2002. [RFC3279] Bassham, L., Polk, W., and R. Housley, "Algorithms and Identifiers for the Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile", RFC 3279, April 2002. [RFC4050] Blake-Wilson, S., Karlinger, G., Kobayashi, T., and Y. Wang, "Using the Elliptic Curve Signature Algorithm (ECDSA) for XML Digital Signatures", RFC 4050, April 2005. [RFC4492] Blake-Wilson, S., Bolyard, N., Gupta, V., Hawk, C., and B. Moeller, "Elliptic Curve Cryptography (ECC) Cipher Suites for Transport Layer Security (TLS)", RFC 4492, May 2006. [SA] T. Satoh, K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comm. Math. Univ. Sancti Pauli 47, 81-92, 1998. [SEC1] Certicom Research. Standards for Efficient Cryptography - SEC 1: Elliptic Curve Cryptography. Version 1.0, 2000. [SEC2] Certicom Research. Standards for Efficient Cryptography - SEC 2: Recomended Elliptic Curve Domain Parameters. Version 1.0, 2000. [Sem] I. A. Semaev. Evaluation of discrete logarithms on some elliptic curves. Math. Comp., 67, 353-356, 1998. [Sma] N. P. Smart. The discrete logarithm problem on elliptic curves of trace one. J. Cryptology 12, 193-196, 1999. 7 Authors' addresses Dr. Manfred Lochter Bundesamt fuer Sicherheit in der Informationstechnik Postfach 200363 53133 Bonn Germany email: manfred.lochter@bsi.bund.de Tel.: +49 228 9582 5643 Dr. Johannes Merkle secunet Security Networks Mergenthaler Allee 77 65760 Eschborn Germany email: johannes.merkle@secunet.com Tel.: +49 6196 95888 55 Fax.: +49 6196 95888 88 Disclaimer of Validity This document and the information contained herein are provided on an "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST AND THE INTERNET ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Copyright Notice Copyright (C) The IETF Trust (2007). This document is subject to the rights, licenses and restrictions contained in BCP 78, and except as set forth therein, the authors retain all their rights.